Bernoulli differential equation

In mathematics, an ordinary differential equation of the form

$y'+ P(x)y = Q(x)y^n\,$

is called a Bernoulli equation when n≠1, 0. It is named after Jacob Bernoulli who discussed it in 1695. Bernoulli equations are special because they are nonlinear differential equations with known exact solutions. A famous special case of the Bernoulli differential equation is the logistic differential equation.

Solution

Let $x_0 \in (a, b)$ and

$\left\{\begin{array}{ll} z: (a,b) \rightarrow (0, \infty)\ ,&\textrm{if}\ \alpha\in \mathbb{R}\setminus\{1,2\},\\ z: (a,b) \rightarrow \mathbb{R}\setminus\{0\}\ ,&\textrm{if}\ \alpha = 2,\\\end{array}\right.$

be a solution of the linear differential equation

$z'(x)=(1-\alpha)P(x)z(x) + (1-\alpha)Q(x).$

Then we have that $y(x) := [z(x)]^{\frac{1}{1-\alpha}}$ is a solution of

$y'(x)= P(x)y(x) + Q(x)y^\alpha(x)\ ,\ y(x_0) = y_0 := [z(x_0)]^{\frac{1}{1-\alpha}}.$

And for every such differential equation, for all $\alpha>0$ we have $y\equiv 0$ as solution for $y_0=0$.

Example

Consider the Bernoulli equation (more specifically Riccati's equation).^[1]

$y' - \frac{2y}{x} = -x^2y^2$

We first notice that $y=0$ is a solution. Division by $y^2$ yields

$y'y^{-2} - \frac{2}{x}y^{-1} = -x^2$

Changing variables gives the equations

$w = \frac{1}{y}$

$w' = \frac{-y'}{y^2}.$

$w' + \frac{2}{x}w = x^2$

which can be solved using the integrating factor

$M(x)= e^{2\int \frac{1}{x}\,dx} = e^{2\ln x} = x^2.$

Multiplying by $M(x)$,

$w'x^2 + 2xw = x^4,\,$

Note that left side is the derivative of $wx^2$. Integrating both sides, with respect to $x$, results in the equations

$\int w'x^2 + 2xw\,dx = \int x^4\,dx$

$wx^2 = \frac{1}{5}x^5 + C$

$\frac{1}{y}x^2 = \frac{1}{5}x^5 + C$

The solution for $y$ is

$y = \frac{x^2}{\frac{1}{5}x^5 + C}$.

References

• Bernoulli, Jacob (1695), "Explicationes, Annotationes & Additiones ad ea, quae in Actis sup. de Curva Elastica, Isochrona Paracentrica, & Velaria, hinc inde memorata, & paratim controversa legundur; ubi de Linea mediarum directionum, alliisque novis", Acta Eruditorum. Cited in Hairer, Nørsett & Wanner (1993).
• Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems, Berlin, New York: Springer-Verlag, ISBN 978-3-540-56670-0.

1. y'-2*y/x=-x^2*y^2, Wolfram Alpha, 01-06-2013

External links

• Bernoulli equation at PlanetMath.org.
• Differential equation at PlanetMath.org.
• Index of differential equations at PlanetMath.org.